Dimension of the set of positive solutions to nonlinear equations and applications

Abstract

We study the covering dimension of the set of (positive) solutions to various classes of nonlinear equations involving condensing and A-proper maps. It is based on the nontriviality of the fixed point index of a certain condensing map or on oddness of a nonlinear map. Applications to nonlinear singular integral equations and to semilinear ordinary and elliptic partial differential equations are given with finite or infinite dimensional null space of the linear part.

Keywords

• Covering dimension, semilinear equations
• fixed point index
• condensing and A-proper maps
• singular integral equations
• exponential dichotomy
• ordinary differential equations
• elliptic PDE's